Lyapunov function search method for analysis of nonlinear systems stability using genetic algorithm

A wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable state vector is considered. The problem of finding a special function (Lyapunov function), which guarantees asymptotic stability for the presented class of nonlinear systems in the framework of the second Lyapunov method, is posed. It is known that the search for the Lyapunov function is an extremely difficult problem that has no universal solution in stability theory. The methods of selection or search of the Lyapunov function for stability analysis of closed linear stationary systems and for nonlinear objects with explicitly expressed linear dynamical and nonlinear static parts are well studied. At the same time, no universal approaches to finding the Lyapunov function for a more general class of nonlinear systems have been identified. In this paper, we propose a new approach to the search of the Lyapunov function for analyzing the stability of smooth continuous dynamic nonlinear systems with a measurable state vector. The essence of the proposed approach consists in the representation of some function through the sum of nonlinear summands representing the elements of the object state vector multiplied by unknown coefficients. The search for these coefficients is performed using a classical genetic algorithm including mutation, selection, and crossover operations. The found coefficients provide all the necessary conditions for the Lyapunov function (within the framework of the second Lyapunov method). The genetic algorithm approach does not require a training sample which imposes restrictions in the form of the structure of control objects included in it. A new method for finding the Lyapunov function represented as a nonlinear series with known functions multiplied by unknown coefficients is proposed. The effectiveness of the proposed method is demonstrated using computer simulations with a fixed number of iterations and varying population size. The dependence of the number of successfully found Lyapunov functions on the number of iterations of the genetic algorithm has been established. The convergence of the genetic algorithm using Holland's schemes is analyzed. It is shown that the values of the sought coefficients of the potential Lyapunov function, at each algorithm iteration, approach the coefficients of the Lyapunov function which was also represented as a Taylor series. The method proposed in this paper outperforms known analogs in terms of speed, considers the decomposition of the potential Lyapunov function into a Taylor series with unknown coefficients, instead of using counterexamples or template functions.